# **Extreme levels**

When designing port infrastructure projects, coastal constructions or oil rigs, it is important to know the minimum and maximum sea level over long periods (up to a century). While maximum tide heights are known, it is vital to examine the likelihood of positive and negative surges. Such studies require long series of good quality measurements.

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## **Definition of positive and negative storm surge - extreme levels**

### **Storm surge**

Instantaneous storm surge is the difference at time t between the observed water level and the predicted water level. A surge is positive if the water level is higher than the expected tide, and negative if lower. Storm surge is mainly meteorological in origin: it is generated during the passage of low pressure systems or anticyclones, by changes in atmospheric pressure and winds. It may also have other origins: waves, seiches, tsunamis, etc.

High tide surge is the difference between the height of the sea observed and the predicted high tide (astronomical tide); this does not necessarily occur at the same time. Similarly, low tide surge is the difference between the low tide observed and the low tide predicted.

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### **Extreme levels**

To study extreme surges, the appropriate variables are high tide surge and low tide surge, not the instantaneous storm surge. By definition these values are free from the effects of the phase difference between observation and prediction. Thus, if only one phase difference exists between the predicted height and observed height either for physical reasons or because the harmonic constants are poorly determined (eg, short-term and poor quality records), the instantaneous storm surge will be even more significant, which is not the case for high tide and low tide surges.

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## **Return period**

As a result, the highest high water level, a component of which is random, is a concept that makes sense only if it is estimated in terms of probability. We must determine the mean interval of time, called the return period, between two rare events with sea levels above a certain threshold.

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### **Determining the return period**

Expressed in this way, the return period would seem to be determined simply by calculating the mean. However, for the mean to be significant, the observation periods must be much longer than the return periods being calculated. Given the available observations, we can hardly estimate return periods longer than two or three years in most cases.

However, it is possible to effectively address this problem for ports, where more than 10 years of tidal observations are available, based on the fact that storm surges and tide are largely independent. If there are many tide observations available, it is easy to calculate the probability distributions governing rare but not exceptional events like high spring tides or major storm surges. But the two types of events may never have been observed simultaneously. However, the return period for this type of very rare event can be calculated with a good degree of confidence by combining the probability distributions for the tide and for surges. The figures show the results for Brest and present the probability for a predicted high tide to be equal to a given value within 1 cm.

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## **Gumbel Distribution**

The problem that arises with storm surges is calculating the probability associated with a storm surge higher than a given value. One difficulty is that very high but rare surges cannot be ignored. Events that have never been observed must be taken into account using an extrapolation model. The model is called "Gumbel Distribution", which is used for estimating river flood peaks. This model was applied to the longest available series of tide measurements (Brest) and the distribution was found to be well suited to the marine environment.

Gumbel distribution results from the study of extreme values of anindependent random variable from the same arbitrary distribution. Extreme value variables were first analysed by Fisher and Tippet and completed by Gumbel. Gumbel distribution is just one of many derived from the theory of probability. It turns out that it is well suited to flooding events, which explains its success.

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## **Results**

Because the tide and surges are addressed separately, the choice of an extrapolation model is not really critical for estimating return periods of extreme levels.

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The figure above shows the probability that storm surge will exceed a certain value, in the coordinate system defined by Gumbel. If the distribution defined by the relation were observed, the experimental points, giving a staircase curve here, would be aligned. The dotted line is the one that best passes through the cluster of experimental points. The two thin lines on either side of the dotted line limit the zone where 90% of the data points should fall if the extrapolation model is selected properly.

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The figure is the result of combining the probability distributions of storm surges and predicted high tides. The area is bounded by the extreme values of predicted high tides.

Here the function gives the probability of observing high tide levels greater than a given value, which can be translated in terms of associated return periods. This example focuses only on the presentation of heights greater than the highest astronomical tide (negative surges are not addressed here).

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## **Mapping the estimated return periods**

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